General solution for differential equation

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Homework Help Overview

The problem involves finding the general solution for a second-order linear homogeneous differential equation of the form y'' + cy' + 6y = 0, where c is a constant. The original poster is uncertain about how to handle the constant c in the context of finding the solution.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the necessity of solving for the roots of the characteristic equation using the quadratic formula and the implications of the discriminant on the nature of the solutions. There is a question about whether c can be set to a specific value for simplification, and some suggest that the general case should be maintained.

Discussion Status

The discussion is ongoing, with participants providing guidance on the need to solve for the roots of the characteristic equation. There is an acknowledgment of the complexity involved in potentially rescaling variables to eliminate c, indicating that multiple approaches are being considered.

Contextual Notes

Participants are grappling with the implications of the constant c in the differential equation and the constraints of finding a general solution without arbitrarily simplifying the problem.

bhsmith
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Homework Statement



y'' + cy' + 6y = 0 (where c is a constant)

I need to find y(t) which i believe is the equation for a general solution.

Homework Equations



It would be r^2 + cr + 6y = 0 then I need to find the roots and create the general solution. My problem is I don't know what to do about the c, can i take the c to be =1 or do I just create a solution with a general c, and if so how would I do that?

The Attempt at a Solution

 
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You'd need to solve for r using the quadratic equation formula and depending on whether the part in the sqrt sign is = 0, > 0 or < 0, would define the general solution.
 
You can't arbitrarily set c to 1. You need to solve the general case. Start by finding the roots to the characteristic equation.
 
If you really want to get rid of c, then you might think about rescaling y and x but that is the complicated solution.
 

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